Optimal. Leaf size=53 \[ -\frac{3 \sin (a+b x) \cos ^{\frac{7}{3}}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(a+b x)\right )}{7 b \sqrt{\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.0125782, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2643} \[ -\frac{3 \sin (a+b x) \cos ^{\frac{7}{3}}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(a+b x)\right )}{7 b \sqrt{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \cos ^{\frac{4}{3}}(a+b x) \, dx &=-\frac{3 \cos ^{\frac{7}{3}}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{7 b \sqrt{\sin ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0584261, size = 53, normalized size = 1. \[ -\frac{3 \sqrt{\sin ^2(a+b x)} \cos ^{\frac{7}{3}}(a+b x) \csc (a+b x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(a+b x)\right )}{7 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( bx+a \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cos \left (b x + a\right )^{\frac{4}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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